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The Journal of Chemical Physics

Entangled trajectory molecular dynamics in multidimensional systems: Two-dimensional quantum tunneling through the Eckart barrier

Lifei Wang, Craig C. Martens, Yujun Zheng

Published Jul 18, 2012• 28 citations
Publisher

Abstract

In this paper, we extend the entangled trajectory molecular dynamics (ETMD) method to multidimensional systems. The integrodifferential form of the evolution equation for the Wigner function is employed, allowing general potentials not represented as a polynomial to be treated. As the example, the method is applied to a two-dimensional model of scattering from an Eckart barrier. The results of ETMD are in good agreement with quantum hydrodynamics and exact quantum simulations. By comparing the quantum and classical trajectory in phase space, the quantum tunneling phenomenon is interpreted vividly.

DOI
10.1063/1.4736559

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In this paper, we extend the entangled trajectory molecular dynamics (ETMD) method to multidimensional systems. The integrodifferential form of the evolution equation for the Wigner function is employed, allowing general potentials not represented as a polynomial to be treated. As the example, the method is applied to a two-dimensional model of scattering from an Eckart barrier. The results of ETMD are in good agreement with quantum hydrodynamics and exact quantum simulations. By comparing the quantum and classical trajectory in phase space, the quantum tunneling phenomenon is interpreted vividly.
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In this Letter, we present a new method for simulating quantum processes in the context of classical molecular dynamics simulations. The approach is based on solving numerically the quantum Liouville equation in the Wigner representation using ensembles of classical trajectories. Quantum effects arise in this formulation as a breakdown of the statistical independence of the ensemble. New interaction forces between ensemble members are derived, which require the trajectory ensemble representing the state to evolve as an entangled, unified whole. The method is applied to the simulation of quantum tunneling in a one-dimensional model system, yielding excellent agreement with exact quantum calculations.
In this paper, we describe a new method for simulating quantum processes using classical-like molecular dynamics. The approach is based on solving the quantum Liouville equation in the Wigner representation using ensembles of classical trajectories in phase space. The nonlocality of quantum mechanics is incorporated in the trajectory representation as nonclassical interactions between the members of the ensemble, leading to an entanglement of their evolution. The statistical independence of the individual trajectories making up an ensemble in the classical limit is lost when quantum effects are included, and the entire state of the system must be propagated as a unified whole. We develop the formalism and its numerical implementation, and illustrate its application on two model problems of quantum mechanical tunneling: escape from a metastable well and wave packet penetration of the Eckhart barrier.
Quantum phase space theory is widely used to investigate physical, and chemical questions. It gives us a powerful tool to study the relations between the classical and quantum world. In quantum phase space theory, entangled molecular dynamics method is developed based on quantum Liouville equation. It is different from classical Hamilton dynamics, the trajectories are entangled i.e. not independent with each other. The quantum effect in the system can be described using the semiclassical theory, and give a vivid physical picture. The quantum tunneling, photodissociation, and self-correlation are investigated using the entangled trajectory. In this paper, we review the recent development in this field.
The quantum Hamilton-Jacobi equation for the action function is approximately solved by propagating individual Lagrangian quantum trajectories in complex-valued phase space. Equations of motion for these trajectories are derived through use of the derivative propagation method (DPM), which leads to a hierarchy of coupled differential equations for the action function and its spatial derivatives along each trajectory. In this study, complex-valued classical trajectories (second order DPM), along which is transported quantum phase information, are used to study low energy barrier transmission for a model two-dimensional system involving either an Eckart or Gaussian barrier along the reaction coordinate coupled to a harmonic oscillator. The arrival time for trajectories to reach the transmitted (product) region is studied. Trajectories launched from an "equal arrival time surface," defined as an isochrone, all reach the real-valued subspace in the transmitted region at the same time. The Rutherford-type diffraction of trajectories around poles in the complex extended Eckart potential energy surface is described. For thin barriers, these poles are close to the real axis and present problems for computing the transmitted density. In contrast, for the Gaussian barrier or the thick Eckart barrier where the poles are further from the real axis, smooth transmitted densities are obtained. Results obtained using higher-order quantum trajectories (third order DPM) are described for both thick and thin barriers, and some issues that arise for thin barriers are examined.
We study a wavepacket tunneling in one-dimensional periodically driven double-well system using entangled trajectory molecular dynamics method. The tunneling dynamics dependents on the amplitude and frequency of the driven force are present. Both resonant and nonresonant tunneling process are enhanced by the driven force when the system is chaotic under classical dynamics. We give entangled trajectory in phase space compared to corresponding classical trajectory with same initial state to visually show quantum tunneling process. The average values of quantum tunneling probability after long time evolution have been shown in the parameter spaces, the effect of resonance and chaos on the tunneling dynamics are present. The relation between chaos and the uncertainly product is discussed in the end. © 2016 Wiley Periodicals, Inc.